On the solution of the two-dimensional ising model with an imaginary magnetic field βH=h=iπ/2

Abstract

The aim of this note is to give a simple and direct proof of the Lee-Yang expression for the solution of the two-dimensional Ising model at exp [-2h] = z = 1, which corresponds to a pure imaginary magnetic field equal to in~2 = h = f iB , fl = ( k T ) L We briefly recall the history of this particular problem in the following way. In one of the two fundamental papers celebrating the circle theorem and the spontaneous magnetization of the Ising model (~) the solution of the Ising model in magnetic field at the point z = 1 was announced. As was pointed out later by BAXTER (2), LEE and YX-~G never gave explicitly the proof of their result for this particular problem, but in (~) it was noted that the solution could be obtained by an extension of the method due to Kxc and WARD for the case z,=, 1 ( h = 0 ) . In a second step (2.3), BXXTER derived the Lee-Yang formula for the free-energy density using the combinatorial approach. This was a consequence of a strong analysis of the group structure which can be used to characterize global properties such as area in the Ising model with magnetic field. In fact, BAXTER introduced a system of weights, which, for any closed path, gives, in addition to (2n) -~ time, the change in the argument of the tangent vector (modulo 2), the number of enclosed units of area (modulo 2) corresponding to z = 1. Nevertheless, in the first step of his proof, BAXTER used an extension of the Sherman theorem on path (z = 1) (as), but no explicit proof was given about validity of this extension for z = 1. In this context, we think therefore, that it is interesting to give another more simple and direct proof of the Yang-Lee fornmla, as an application of another concept of path described by KADANOFF and CEVA (e) and used for the duality transformation of correlations (7). In a second step we make use of the result of UTIYAMA for the generalized anisotropic Ising model on a square latt ice in zero field.